# 隱函數

$\text{e} = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$

## 例子

### 反函數

x = f(y)

x表示y。這個解是

$y = f^{-1}(x).$

$R(x,y) = x-f(y) = 0. \,$

1. 對數函數 ln(x) 給出方程xey = 0或等價的x = ey的解 y = ln(x) . 這裡 f(y) = ey 並且 f−1(x) = ln(x).
2. The product log is an implicit function giving the solution for y of the equation xy ey = 0.

### 代數函數

$a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 \,$

where the coefficients ai(x) are polynomial functions of x. Algebraic functions play an important role in mathematical analysis and algebraic geometry. A simple example of an algebraic function is given by the unit circle equation:

$x^2+y^2-1=0. \,$

Solving for y gives an explicit solution:

$y=\pm\sqrt{1-x^2}. \,$

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation.

While explicit solutions can be found for equations that are quadratic, cubic, and quartic in y, the same is not in general true for quintic and higher degree equations, such as

$y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0. \,$

Nevertheless, one can still refer to the implicit solution y = g(x) involving the multi-valued implicit function g.

## 隱函數的導數

• 隱函數左右兩邊對$x$求導（但要注意把$y$看作$x$的函數）；
• 利用一階微分形式不變的性質分別對$x$$y$求導，再通過移項求得$\frac {dy}{dx}$的值；
• 把n元隱函數看作(n+1)元函數，通過多元函數偏導數的商求得n元隱函數的導數。舉個例子，若欲求$z=f(x,y)$的導數$\frac {dy}{dx}$，那麼可以將原隱函數通過移項化為$f(x,y,z)=0$的形式，然後通過$\frac {dy}{dx} = -\frac{F'_x}{F'_y}$（式中$F'_y$$F'_x$分別表示$y$$x$$z$的偏導數）來求解。

### 示例

• 針對$y^n$

$\frac{d}{dx}y^n = n \cdot y^{n-1}\frac{dy}{dx}$

• 針對$x^m y^n$

$\frac{d}{dx}x^m y^n = n \cdot x^m y^{n-1}\frac{dy}{dx} + m \cdot x^{m-1} y^n$

• $\ 12x^7-7x^4 y^3+6xy^5-14y^6+25=10$對x的導數。

${\color{Blue}12x^7}{\color{Red}-7x^4 y^3}{\color{Green}+6xy^5}{\color{Brown}-14y^6}+25=10$

1.兩邊皆取其相應的導數，得出

${\color{Blue}12\cdot7x^6}{\color{Red}-7\left(3x^4 y^2\frac{dy}{dx} + 4x^3 y^3 \right)}{\color{Green}+6\left(5xy^4\frac{dy}{dx} + y^5\right)}{\color{Brown}-14\cdot 6y^5\frac{dy}{dx}}+0=0$

2.移項處理。

${\color{Blue}84x^6}{\color{Red}- 28x^3 y^3}{\color{Green}+ 6y^5}={\color{Red}21x^4 y^2\frac{dy}{dx}}{\color{Green}- 30xy^4\frac{dy}{dx}}{\color{Brown}+84y^5\frac{dy}{dx}}$

3.抽出導數因子。

${\color{Blue}84x^6}{\color{Red}- 28x^3 y^3}{\color{Green}+ 6y^5}=\left({\color{Red}21x^4 y^2}{\color{Green}- 30xy^4}{\color{Brown}+84y^5} \right)\left( \frac{dy}{dx} \right)$

4.移項處理。

$\frac{dy}{dx} = \frac{{\color{Blue}84x^6}{\color{Red}- 28x^3 y^3}{\color{Green}+ 6y^5}}{{\color{Red}21x^4 y^2}{\color{Green}- 30xy^4}{\color{Brown}+84y^5}}$

5.完成。得出其導數為$\frac{84x^6 - 28x^3 y^3 + 6y^5}{21x^4 y^2 - 30xy^4 + 84y^5}$

6.選擇性步驟：因式分解處理。

$\frac{dy}{dx} = \frac{2\left(42x^6 - 14x^3 y^3 + 3y^5 \right)}{3y^2\left(7x^4 - 10xy^2 + 28y^3\right)}$